Reducible Dehn Surgeries, Ribbon Concordance, and Satellite Knots

Abstract

In this thesis we investigate knots and surfaces in $3$- and $4$-manifolds from the perspective of Heegaard Floer homology, knot Floer homology and Khovanov homology. We first investigate the \emph{Cabling Conjecture}, which states that the only knots that admit reducible Dehn surgeries are cabled knots. We study this question and related conjectures in Chapter \ref{reduciblesurgeries} and develop a lower bound on the slice genus of knots that admit reducible surgeries in terms of the surgery parameters and study when a slope on an almost L-space knot is a reducing slope. In particular, we show that when $g(K)$ is odd and $>3$, the only possible reducing slope on an almost L-space knot is $g(K)$ and in that case the complement of an almost L-space knot does not contain any punctured projective planes. In Chapter \ref{chaptersatellite} we investigate the effect of satellite operations on knot Floer homology using techniques from bordered Floer homology \cite{LOT} and the immersed curve reformulation \cites{HRW,Chen, chenhanselman}. In particular we study the functions $n \mapsto g(P_n(K)), \epsilon(P_n(K))$ and $\tau(P_n(K))$ for some families of $(1,1)$ patterns $P$ from the immersed curve perspective. We also consider the function $n \mapsto \dim(\HFKhat(S^3,P_n(K),g(P_n(K)))$, and use this together with the fibered detection property of knot Floer homology \cite{Nifibered} to determine, for a given pattern $P$, for which $n \in \Z$ the twisted pattern $P_n$ is fibered in the solid torus. In Chapter \ref{Chapterribbon} we answer positively a question posed by Lipshitz and Sarkar about the existence of Steenrod operations on the Khovanov homology of prime knots \cite[Question 3]{MR3966803}. The proof relies on a construction of a particular type of surface, called a ribbon concordance in $S^3 \times I$, interpolating between any given knot and a prime knot together with the fact that the maps induced on Khovanov homology by ribbon concordances are split injections \cite{MR3122052,MR4041014}.